Method for determining the characteristics of crude oils and mixtures of chain molecules by diffusion, relaxation and density measurements

ABSTRACT

The diffusion coefficients and relaxation times of mixtures of alkanes follow simple scaling laws based on the chain length of the constituents and the mean chain length of the mixture. These scaling laws are used to determine chain sizes in a mixture from the distribution of the diffusion coefficients. These scaling laws can be used to determine the mean chain lengths (or chain lengths) of a sample (alkanes or mixtures of alkanes) and therefore the constituents of the sample.

The present invention is a divisional filing of pending U.S. patentapplication Ser. No. 10/864,124, filed Jun. 9, 2004.

FIELD OF THE INVENTION

The present invention relates to a method of modeling alkanes and, moreparticularly, to a method of determining the constituents of an oilmixture and its viscosity.

BACKGROUND OF THE INVENTION

It is well known that the self-diffusion coefficient of a molecule isrelated in some way to its size. For a hard sphere in a fluid withviscosity η_(s), the relationship is given by the Einstein-Stokesequation,

$\begin{matrix}{D = \frac{k_{B}T}{6\pi \; \eta_{s}r}} & (1)\end{matrix}$

where r is the radius of the sphere and η_(s) is the viscosity of thesolvent. This equation suggests that in a mixture with molecules ofdifferent radii, the diffusion coefficient D_(i) of the i^(th) componentis

$\begin{matrix}{D_{i} = \frac{k_{B}T}{6\pi \; \eta_{s}r_{i}}} & (2)\end{matrix}$

where r_(i) is the radius of the i^(th) component. From this, it can beconcluded that the ratio of the diffusion coefficients of any twocomponents in the mixture will depend only on the ratios of the sizes ofthe two molecules and is independent of any other properties of thefluid, such as its viscosity or temperature. Alternatively, Equation (2)implies that for a particular mixture, D_(i)r_(i) is constant for allcomponents in the mixture. In addition, Equation (2) states that thereis a fixed relationship between the diffusion coefficients and theviscosity of the fluid (D∝1/η_(s)).

An application of the hard sphere model to oils may be found in Freedmanet al. “A New NMR Method of Fluid Characterization in Reservoir Rocks:Experimental Confirmation and Simulation Results,” paper SPE 63214presented at the 2000 SPE Annual Technical Conference and Exhibition,Dallas, 1-4 October; Lo et al. “Relaxation Time and DiffusionMeasurements of Methane and Decane Mixtures,” The Log Analyst(November-December 1998); and Lo et al. “Correlations of NMR RelaxationTime with Viscosity, Diffusivity, and Gas/Oil Ratio ofMethane/Hydrocarbon Mixtures,” in Proceedings of the 2000 AnnualTechnical Conference and Exhibition, Society of Petroleum Engineers(October 2000). These articles are hereby incorporated by referenceherein in their entireties.

As shown in FIG. 1, diffusion coefficient D is related to the size ofthe molecule. Diffusion depends on the constituents (components andrelative amounts) of the mixture under analysis. However, existingmodels do not adequately describe the physics of the system sufficientto identify the components of the mixture to an acceptable degree ofcertainty.

However, the hard sphere model is not adequate for describing morecomplicated molecules, such as oils, which are floppy chains. Oneexample of the failure of the hard sphere model is evidenced by themeasurements of diffusion and viscosity in alkanes and oils. In plots oflog D versus log kT/η (see FIG. 2), the data all lie on a single line,regardless of the molecule's radii. These plots are in disagreement withEquation (2), which would imply that the slope depends on the radius ofthe molecule.

As will be shown below, elevated temperature and pressure may influencethe modeling of complicated oils. Applications of the hard sphere modelto oils do not adequately account for the effects of temperature andpressure on the diffusion coefficients and relaxation times.

Accordingly, it is one object of the present invention to provide amethod to more appropriately model oils and oil mixtures.

It is another object of the present invention to provide a model thataccounts for the temperature and pressure dependence of the diffusioncoefficient and relaxation times over a wide range of temperatures andpressures.

SUMMARY OF THE INVENTION

Accordingly, the present inventor has discovered that diffusioncoefficients and relaxation times of mixtures of alkanes follow simplescaling laws based on the chain length of the constituents and the meanchain length of the mixture. These scaling laws are used to determinechain sizes in a mixture from the distribution of the diffusioncoefficients. These scaling laws can be used to determine the mean chainlengths (or chain lengths) in live oils as well as the viscosity of amixture.

It is noted the use of scaling laws have not been used to give acomplete description for mixtures of short chain molecules, crude oils,or the like. As is commonly known in prior art, scaling laws generallyrefer to power laws, where one quantity, x, is equal to anotherquantity, y, raised to some power ν, so that x=ay^(ν). Although ν issometimes determined a priori, often ν and the constant ofproportionality, a, must be determined for the specific properties andconditions of interest. Scaling laws generally describe very largesystems, such as very long chains, and in prior art are not usuallyexpected to be valid for short chains.

In addition, the present invention characterizes the relationshipbetween diffusion coefficients D_(i) or nuclear magnetic relaxationtimes T_(1i) and T_(2i) and molecular composition for mixtures ofalkanes at elevated pressures and temperatures. Using properties of thefree volume theory and the behavior of the density of alkanes, for alarge range of pressures, the diffusion coefficients and relaxationtimes depend on pressure and mean chain length of the mixture onlythrough the density. Accordingly, a method is provided for determiningthe relationship between D_(i), T_(1i) or T_(2i) and composition atelevated pressures, as long as the density of the fluid is known. Inaddition, the relationship between D_(i), T_(1i) or T_(2i) andcomposition can be determined at arbitrary pressure P as long as it isknown at a suitable reference pressure P₀. The scaling laws for D_(i),T_(1i) and T_(2i) and the Ahrrenius dependence on temperature arecombined to obtain the temperature dependence of the diffusioncoefficient and relaxation times. Once the relationship between D_(i),T_(1i) or T_(2i) and composition is known, measurements of the diffusioncoefficients and relaxation times, using various nuclear magneticresonance (NMR) tools such as Schlumberger's MRX™, can be used to obtainthe composition of mixtures of alkanes. In addition, this technique maybe applied to diffusion and relaxation data collected from fluidsampling tools and may be practiced in the field (i.e. downhole) or in alaboratory. In addition diffusion-edited data such as that collectedusing the techniques of commonly owned U.S. Pat. No. 6,570,382 may beevaluated using this method. It is further noted that this technique maybe applied to non-NMR diffusion data as may be known in the art. It isfurther noted that diffusion measurements may be preferred overrelaxation measurements, particularly for small chain lengths (i.e.,methane and ethane) where additional physics come into play.

Accordingly, expressions for diffusion coefficient and relaxation timesmay be determined as a function of chain lengths at elevated pressuresand temperatures using: (1) density data for pure alkanes at the desiredpressures and temperatures and (2) data on diffusion coefficients orrelaxation times for pure or mixed alkanes at one reference pressure andseveral temperatures. Preferably, this data spans the range of chainlengths or densities of interest. Once the relation between D, T₁, or T₂and chain lengths is known, it can be used with the modeling methodsherein to determine chain length distributions from diffusion orrelaxation measurements.

This method is applicable for a wide range of temperatures and pressuresand for chain lengths less than the entanglement length. For pressuresabove about 100 MPa careful selection of the reference pressure isrecommended. It is also noted that for pressures above about 100 MPa theslope of the calibration curve may begin to change and should beaccounted for. If asphaltenes or a large amount of aromatics areexpected to be present, it may be preferable to obtain one or moreadditional measurements to determine the type of molecules in themixtures. A difference between the chain length distribution found bymeasuring the distributions of diffusion coefficients and relaxationtimes may identify the presence of asphaltenes. Asphaltenes form largeaggregates that tumble slowly, T₂ is sensitive to this slow motion whileT₁ is not sensitive to this motion. This leads to a shortening of T₂ ascompared to T₁. This difference between T₁ and T₂ can be used toidentify the presence of asphaltenes.

The method of the present invention can be quite useful in detectinggradients in composition along a well or between wells. If the oil is ofthe type that varies with temperature and pressure as in the alkanemodel, then the NMR derived distribution could be calibrated withlaboratory oil measurements at a few places, and points between themeasured NMR distributions would indicate composition gradients. In theabsence of lab measurements, the NMR distributions can show compositiongradients; however, lab measurements would verify that the temperatureand pressure dependence is of the expected form.

One advantage of obtaining the composition from the diffusion orrelaxation measurements is that it is complementary to opticalmeasurements, first, because they measure different types of physics,and second, because they NMR measurement can give some detailedinformation about the molecules with longer chain lengths, while opticalmeasurements can give details about the exact methane content and thepresence of other gases. Methane can affect the scaling law for the NMRrelaxation times, while the presence of other gases such as CO₂ andnitrogen can affect the density, diffusion, and relaxation. It may beuseful to know how much of these gases are present to properly invertthe T₁ and diffusion data for the chain length distributions. On theother hand, the presence of the large molecules, which is given by theNMR measurements, can greatly affect the properties of the oil, such aswhether it can become waxy.

Because the NMR measurements are sensitive to larger particles, it canalso be useful for detecting phase changes. As waxes or asphaltenesstart to aggregate or precipitate, they should appear in the chainlength distribution as much larger particles, which can signal a phasechange as the temperature or pressure is changed.

Accordingly, the present invention provides a method of determining theconstituents of an oil mixture and its viscosity. The method of thepresent invention includes using commonly known nuclear magneticresonance techniques to determine the diffusion distribution of amixture and, using polymer models, correlating this diffusiondistribution to chain length of the constituents, the mean chain lengthof the mixture, or its viscosity.

Accordingly, a first embodiment of the present invention is a method fordetermining the characteristics of a fluid sample, comprising: (a)obtaining measurements (diffusion or relaxation measurements) on aplurality of calibration samples having one or more known constituents;and (b) determining the scaling law of said plurality of fluid samplesusing as a function of chain length said measurements of (a). To createan accurate calibration, the calibration samples should have a varietyof mean chain lengths. In addition, the calibration samples may be purealkanes or mixtures of alkanes, or a combination thereof. Once thiscalibration is determined, the constituents of a sample underinvestigation may be determined by obtaining either diffusionmeasurements and relaxation measurements, depending on the measurementsmade in (a) above and then applying the scaling law of (b) to thesemeasurements. It is noted that the calibration does not need to beredone for each sample; once a calibration is performed, it may bereused for other samples. In one application of this embodiment, thecalibration measurements and the sample measurements are performed at afirst temperature and a reference pressure. The scaling law may beobtained by performing a two-parameter fit of the function, such as byidentifying the slope and intercept of the scaling law. It is preferableto perform the calibration at the temperature and pressure approximatelyequal to the expected temperature and a second pressure. This methodallows the determination of the mean chain length and the distributionof chain lengths of the constituents of the sample under investigation.From this information, the composition of the sample may be determined.

In a second embodiment, a method for determining the characteristics ofa fluid sample is disclosed, wherein the calibration samples are subjectto different temperatures and the reference pressure. In this case thescaling law becomes a function of mean chain length and temperature andit is no longer necessary to substantially match the temperature of thecalibration samples to the expected temperature of the sample underinvestigation. Now the scaling law may be determined using a fourparameter fit, as described in more detail below.

In a third embodiment, a method for determining the characteristics of afluid sample is disclosed, comprising: (a) obtaining measurements(diffusion or relaxation measurements) of a plurality of calibrationsamples at a first temperature and a reference pressure, wherein thecalibration samples have differing mean chain lengths; (b) determiningthe density of more than one pure alkane or mixtures of alkanes (notnecessarily the same as the calibration samples) at the firsttemperature and the reference pressure, wherein density is determined asa function of mean chain length; (c) obtaining measurements (diffusionor relaxation measurements) of the sample under investigation at thefirst temperature and a second pressure; (d) determining the density ofthe sample under investigation at the first temperature and the secondpressure; (e) applying the density function of (c) to the densitymeasurements of (d) and using the measurements of (a) to determine thescaling law at the second pressure in terms of chain length; (f)applying the scaling law of (e) to the data of (c) to determine thecomposition of the sample under investigation. The density measurementsof (b) may be obtained from standard look-up tables (such as the NISTwebbook). This method can be used to determine the composition of thesample under investigation by determining the distribution of chainlengths of the constituents of the sample under investigation. It isnoted that the density measurements of (b) can be any density,including, but not limited to, mass density, carbon density, andhydrogen density (more commonly known as the hydrogen index).

The fourth embodiment comprises a manipulation of the third embodiment,wherein a range of temperatures is accounted for. More specifically, amethod for determining the characteristics of a fluid sample isdisclosed, comprising: (a) obtaining measurements (diffusion orrelaxation measurements) of a plurality of calibration samples atreference pressure and at more than one temperature, wherein thecalibration samples have differing mean chain lengths; (b) determiningthe density of more than one pure alkane or mixtures of alkanes at thereference pressure and at a temperature within or near the range oftemperatures in (a), wherein density is determined as a function of meanchain length; (c) obtaining measurements (diffusion or relaxationmeasurements) of the sample under investigation at a second pressure andat a temperature within or near the range of temperatures in (a); (c)determining the density of the sample under investigation at the secondpressure and at a temperature within or near the range of temperaturesin (a); (d) applying the density function of (c) to the densitymeasurements of (d) and using the measurements of (a) to determine thescaling law at the second pressure in terms of chain length; (e)applying the scaling law of (e) to the data of (c) to determine thecomposition of the sample under investigation.

In a fifth embodiment, a method for determining the characteristics of afluid sample is disclosed, comprising: (a) obtaining measurements(diffusion or relaxation measurements) of a plurality of calibrationsamples at a first temperature and a reference pressure, wherein thecalibration samples have differing mean chain lengths; (b) obtainingmeasurements (diffusion or relaxation measurements) of a sample underinvestigation at the first temperature and at a second pressure; (c)determining the relationship of volume of one or more alkanes ormixtures of alkanes (not necessarily the calibration sample) to (i) themean chain length at the first temperature and the reference pressureand (ii) the mean chain length at the first temperature and a secondpressure; (d) determining the scaling law as a function of chain length,using the functions of (c) and the measurements of (a); (e) applying thescaling law of (d) with the measurements of (b) to determine thecomposition of the sample under investigation. The volumes of (c) may beobtained using standard look-up tables (such as the NIST webbook).Further, the volumes may be any volume, including, but not limited to,volume per mole (molar volume), volume per hydrogen atom, or volume percarbon atom.

The sixth embodiment is a manipulation of the fifth embodiment toaccount for various temperatures. More specifically, a method fordetermining the characteristics of a fluid sample, comprising: (a)obtaining measurements (diffusion or relaxation measurements) of aplurality of calibration samples at more than one temperature and areference pressure, wherein the calibration samples have differing meanchain lengths; (b) obtaining measurements (diffusion or relaxationmeasurements) of a sample under investigation at a temperature within ornear the range of temperatures of the measurements of (a) and at asecond pressure; (c) determining the relationship of volume of alkanesor mixtures of alkanes to (i) the mean chain length at the referencepressure and (ii) the mean chain length at the second pressure; (d)determining the scaling law in terms chain length and temperature at thesecond pressure using the functions of (c) and the measurements of (a);(e) applying the scaling law of (d) with the measurements of (b) todetermine the composition of the sample under investigation.

It is envisioned that these methods may be performed in a laboratory orat the point of sampling. For example, these methods may be particularlyuseful in the characterization of oilfields and may be used on samplesobtained from the earth formation or within the earth formation.

Further features and applications of the present invention will becomemore readily apparent from the figures and detailed description thatfollows.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a graph showing the relationship between diffusion coefficient(D) and the size of the molecule.

FIG. 2 is a graph showing the relationship between log D and log η/T totest the hard sphere model.

FIG. 3 is a graph showing the relationship between diffusion coefficientand chain length for pure alkanes and mixtures.

FIG. 4 illustrates the Rouse model.

FIG. 5 is a graph demonstrating that D_(i)N_(i) ^(ν) and D_(g)r_(g) areconstant in binary and ternary mixtures.

FIG. 6 is a model showing the volume at each segment and the extra freevolume at each edge.

FIG. 7 is a graph depicting scaled diffusion coefficients for purealkanes and mixtures as a function of mean chain length.

FIG. 8 is a graph depicting the scaling law for live oils.

FIGS. 9( a)-(d) are graphs showing chain length distributions for twocrude oils.

FIG. 10 is a graph showing measured viscosity versus calculatedviscosity for pure alkanes.

FIG. 11 is a graph showing relaxation times T₁ versus density for C8,C10, C12, and C16 at elevated pressures.

FIG. 12 is a graph showing scaled relaxation times (N^(k)T₁) versusdensity for C8, C10, C12, and C16.

FIG. 13 is a graph showing scaled relaxation times N_(i) ^(k)T_(1i)versus density for pure alkanes and mixtures at T=25° C.

FIG. 14 is a graph of scaled diffusion coefficients N_(i) ^(ν)D_(i)versus density at T=25° C.

FIG. 15 is a graph of scaled relaxation times N_(i) ^(k)T_(1i) versuseffective chain length N_(eff).

FIG. 16 is a graph of molar volume ν_(T) versus mean chain length Nusing the data of Zega's PhD thesis (see below).

FIG. 17 is a graph of molar volume ν_(T) versus mean chain length N fordead and live alkanes at 50° C.

FIGS. 18( a)-(f) are graphs of molar volume ν_(T) versus mean chainlength N for dead and live pure alkanes and mixtures of alkanes at avariety of pressures and temperatures.

FIG. 19 is a graph of scaled diffusion coefficient N_(i) ^(ν)D_(i)versus mean chain length N at 25° C.

FIG. 20 is a graph of scaled diffusion coefficient(ν_(e)(P₀)/ν_(e)(P))^(β)r_(i)D_(i) versus mean chain length+1 ( N+1) fordead and live alkanes at 30° C.

FIG. 21 is a graph of diffusion coefficient D versus reciprocaltemperature for pure alkanes at atmospheric pressure.

FIG. 22 is a graph of scaled relaxation time T₁N^(k) for effective chainlength N_(eff).

DETAILED DESCRIPTION OF THE INVENTION Modeling Oils

Properties that follow from Equation (2) suggest a method for fluidtyping using diffusion data of oils. The ratio of diffusion coefficientswithin a given mixture gives the ratio of the “sizes” of the components.Furthermore, if it is known how the product D_(i)r_(i) depends on theparticular mixture, then the actual sizes of the molecules may berecovered, not just their ratios. As will be discussed below, polymerphysics concepts may be used to show how diffusion data from mixtures ofalkanes and live alkanes can be used to recover information about thechain lengths within a mixture as well as information about the meanchain length. The polymer model can also be used to explain therelationship between diffusion and viscosity as shown in FIG. 2.

FIG. 3 shows that as a function of chain length, the diffusioncoefficient can vary greatly. This graph is based on data collected fromnumerous sources, including Dymond et al. “The Temperature and DensityDependence of the Self-Diffusion Coefficient of n-hexadecane,” Molec.Phys. 75(2):461-466 (1992) and Marbach et al. “Self- and MutualDiffusion Coefficients of some n-Alkanes at Elevated Temperature andPressures,” Z. Phys. Chem., Bd. 193, S:19-40 (1996) (incorporated byreference herein in their entireties). The data shown is measured atdiffering temperatures and pressures for differing mixtures.Accordingly, diffusion coefficients also vary as a function oftemperature, pressure, and constituents of the mixture. Thus, animproved model for diffusion in mixtures is required, if one is able touse the diffusion coefficients to invert for chain lengths. As shownbelow, the ratio of diffusion coefficients in a mixture depends only onthe relative size of the molecules in the mixture and the internalviscosity depends only on the mean chain length of the mixture.

An aspect of this invention is to show how polymer models may be used todescribe mixtures of alkanes. Doi et al. The Theory of Polymer Dynamics,Oxford University Press, New York (1996) and Ferry, ViscoelasticProperties of Polymers, John Wiley & Sons, Inc., New York (1980)(incorporated by reference herein in their entireties) provides acomplete description of the polymer models. In the polymer models, themolecule is modeled by a chain of beads with a gaussian distribution ofbond lengths between adjacent beads, which leads to a spring-likeinteraction between adjacent beads (also referred to as a “gaussianchain”). Each bead is subject to a Brownian force which is due to (1)the solvent molecules if the polymer is in a solvent and (2) the beadsof the other polymers if the polymer is in a melt. For long chains(roughly 100 beads or longer), this model is a good description,provided an effective distance between adjacent beads, l, is assignedthat is not always equal to the actual distance between the beads. Inthat case, the mean radius of gyration of the chain is given by

$\begin{matrix}{R_{g} = \frac{N^{D}l}{\sqrt{6}}} & (3)\end{matrix}$

where N is the chain length. For alkanes, N is equal to the number ofcarbon atoms. The exponent ν is approximately equal to ½ if there are noexcluded volume effects and ⅗ if there are excluded volume effects.

For alkanes found in oils, the chain lengths are usually too short to beperfectly described by the ideal gaussian chain, because the chains arestiffer than a true gaussian chain. For very long alkanes andpolymethylene chains (with N≧100) where gaussian behavior is observed,the parameter l is about √{square root over (6.7)} times the actualdistance between carbon atoms. For shorter chains, l is not a constant.Instead, it varies with chain length, decreasing to the actual distancebetween carbon atoms for a chain length of two. However, even when othertypes of interactions between beads are used, many of the results forgaussian chains still apply, at least qualitatively.

Oils and liquid alkanes are melts. In melts, it is usually assumed thatthe hydrodynamic effects are screened out by the chains and that theexcluded volume effects within a chain are balanced out by the excludedvolume effects between differing chains. In that case, the melt isdescribed by the Rouse model (shown in FIG. 4), which is the simplestversion of the polymer models (see Rouse, J. Chem. Phys. 21:1272 (1953)(incorporated by reference herein in its entirety)). In the Rouse model,the translational diffusion coefficient of the molecule is given by

$\begin{matrix}{D = \frac{k_{B}T}{N\; \xi}} & (4)\end{matrix}$

where ξ is the friction constant for a bead and N is the number of beadsin the chain. The Rouse model accounts for the gaussian interactionbetween nearest neighbors and each bead feeling the coefficient offriction, ξ, from the surrounding fluid.

For the shorter chains such as the alkanes, the hydrodynamic effects arenot necessarily all screened out. The Zimm model adds the hydrodynamiceffects to the Rouse model (see Zimm, J. Chem. Phys. 24:269 (1956)(incorporated by reference herein in its entirety)). In the Zimm model,the translational diffusion coefficient is given by

$\begin{matrix}{D \approx {0.2\frac{k_{B}T}{\eta_{S}\sqrt{6}R_{g}}}} & (5)\end{matrix}$

and, in the absence of excluded volume effects, the rotational diffusioncoefficient is given by

$\begin{matrix}{D_{R} = \frac{\sqrt{\left( {3\pi} \right)}k_{B}T}{{\eta_{s}\left( {\sqrt{6}R_{g}} \right)}^{3}}} & (6)\end{matrix}$

where the radius of gyration is given by Equation (3). The viscosityη_(s) is the viscosity that each bead sees. For example, if the polymeris dissolved in a solvent, it is the viscosity of the solvent. In themelt, it may be considered as the “internal viscosity” that each beadsees due to the high frequency motion of all the beads on the otherchains in the melt. Equations (5) and (6) have a form very similar tothe diffusion coefficients for the hard sphere model, but now the radiusof the molecule scales with chain length as a function of N^(1/3) (ascompared to Equation (3) where the radius scales with chain lengthN^(ν)), and the internal viscosity η_(s) is not necessarily equal to themacroscopic viscosity of the fluid.

Even though the chains encountered in most oils are short and thereforeare not perfect gaussian chains, many ideas about the collective motionof the internal degrees of freedom inherent in polymer models stillapply. As discussed below, many of the polymer results appear to applyto the alkanes, at least qualitatively, if not quantitatively.Accordingly, the chains follow similar scaling laws.

Based on the polymer models, it is expected that the diffusioncoefficient of the i^(th) component of an alkane in a mixture is

$\begin{matrix}{D_{i} = \frac{{ak}_{B}T}{N_{i}^{\upsilon}\eta_{0}}} & (7)\end{matrix}$

where N_(i) is the chain length or number of carbon atoms of the alkane;η₀ is the internal viscosity or coefficient of friction associated withthe motion of a single bead; and a is a constant that depends on whichmodel is appropriate for describing the fluid.

According to the various polymer models, it is expected that 0.5≦ν≦1. Inthe example presented herein, ν≈0.7 (see the fit of the data in FIG. 5),in contrast to the value of ν=1 which is usually assumed for polymer andalkane melts. The lower value of ν probably reflects the fact that thehydrodynamic effects are not fully screened in the alkane melts and thatexcluded volume effects or the stiffness of the chains play a role. Itis still expected that for very long chains (but before the onset ofentanglements) the value of ν will approach the ideal value of 1.

The methane and ethane molecules (and, most likely, the alkanes up toand including pentane) are more appropriately described by the hardsphere model, so it is expected that their diffusion coefficients willhave the form

$\begin{matrix}{D_{i} = \frac{{ak}_{B}T}{r_{i}\eta_{0}}} & (8)\end{matrix}$

where r_(i) is related to the radius of the molecule. In this equation,the same constant a and internal viscosity η₀ is used as in Equation(7). This means that r_(i) is unitless and thus can be compared with thevalue of N_(i) ^(ν) appropriate for the longer chains. Thus, one aspectof this invention is treating the oil and dissolved gas in a similarmanner so that for both the oil and the gas the diffusion equation canbe given by Equation (8). Fitting to the data for binary mixtures withethane and methane in Helbaek et al., “Self-Diffusion Coefficients ofMethane or Ethane Mixtures with Hydrocarbons at High Pressure by NMR,”J. Chem. Eng. Data 41:598-603 (1996) (incorporated by reference hereinin its entirety) gives, for methane

r_(m)≈1.64  (9)

and for ethane

r_(e)≈2.32  (10)

Within a mixture, then, the ratio of the diffusion coefficients of anytwo components depends on the ratio of their radius or chain length tosome power. In other words, if components 1 and 2 are oils, then theirdiffusion coefficients have the ratio

D ₁ /D ₂ =N ₂ ^(ν) /N _(i) ^(ν)  (11)

regardless of any other properties of the mixture, such as itscomposition, temperature or pressure. Similarly, for a gas and an oil,the ratio would be

D _(g) /D _(o) =N _(o) ^(ν) /r _(g)  (12)

These equations are similar to what one would expect reading Bearman,“Statistical Mechanical Theory of Diffusion Coefficients in BinaryLiquid Solutions,” J. Chem. Phys. 32(5):1308-1313 (1959) (incorporatedby reference herein in its entirety) for nearly ideal fluids where themolar volume of the fluids does not change much upon mixing. However, inthe present case, the ratio depends on the effective radii of themolecules r_(i) or on N^(ν) instead of on the molar volume of thefluids. Equations (11) and (12) imply that by knowing the ratios of thediffusion coefficients in a mixture, the ratios of the radii or chainlengths of the compositions may be determined.

Expressing Equations (11) and (12) slightly differently, it is expectedthat D_(i)N_(i) ^(ν) and D_(i)r_(i) are constant for all componentswithin a particular mixture, as shown in FIG. 5. For live oils, thescaled diffusion coefficient of the gas component D_(g)r_(g) is plottedalong the x-axis. For the dead mixtures, the scaled diffusioncoefficient of the lighter oil D_(l)N₁ ^(ν) is plotted along the x-axisinstead. The scaled diffusion coefficient of the heavier oil componentis plotted along the y-axis. The theoretical prediction is shown by theblack line. The pressures range from atmospheric pressure to 60 MPa andthe temperature ranges from 25 to 60° C. The diffusion coefficients arein units of 10⁻⁵ cm²/s. All the points lie very close to the solid linepredicted by Equations (11) and (12). With the values of the effectiveradii and exponent ν given above (where ν=0.7; see Equations (9) and(10)), the fit works very well both for the live oils and ternarymixtures, in addition to the binary mixtures of liquid alkanes. This fitis noteworthy because the live oils are not ideal fluids. For example,the data on the density of methane and decane mixtures in Lee et al.“Viscosity of Methane-n-Decane Mixtures,” J. Chem. Phys. 11(3):281-287(1966) (incorporated by reference herein in its entirety) clearly showsthat the volumes are not strictly additive (i.e., the excess volume isnot negligible). Also, some of the mixtures contain benzene or squalene,neither of which are linear. However, they both appear to behave verysimilarly to what would be expected if they were linear chains. Althoughthe data in FIG. 5 is from the literature, the plot in FIG. 5 representsa new scientific result which explains part of this invention. It showsthat the relations in Equations (7) and (8) are valid within mixtures ofalkanes (and not just for hard spheres or polymers) over a wide range oftemperatures and pressures and determines the values of ν and r_(i).

For the alkanes, to a good approximation, the products D_(i)N_(i) ^(ν)and D_(i)r_(i) depend only on the mean chain length in the mixture.Thus,

$\begin{matrix}{{D_{i}N_{i}^{\upsilon}} = {{D_{i}r_{i}} = {\frac{{ak}_{B}T}{\eta_{0}} = {g\left( \overset{\_}{N} \right)}}}} & (13)\end{matrix}$

where N is the average chain length in the mixture, given by

$\begin{matrix}{\overset{\_}{N} = {\sum\limits_{i}\; {x_{i}N_{i}}}} & (14)\end{matrix}$

and x_(i) is the mole fraction of the i^(th) component in the mixture.The function g( N) also depends on the temperature and pressure. Aversion of this property, with different values for ν, was originallysuggested by Van Geet et al. based on diffusion in mixtures of C8, C12,and C18.

The free volume model for alkanes in von Meerwall et al. “Diffusion ofLiquid n-Alkanes: Free-Volume and Density Effects,” J. Chem. Phys.108(10):4299-4304 (1998) and von Meerwall et al. “Diffusion in BinaryLiquid n-Alkane and Alkane-polyethylene Blends,” J. Chem. Phys.111(2):750-757 (1999) (incorporated by reference herein in theirentireties) can be used to give an explanation for this property. Inaddition to depending on the activation energy E_(a) for hopping, theprobability that segments of the chain will hop depends on whether thereis enough free volume available for them to move into. According to thisfree volume picture, for a pure fluid the function g(N) has the form

$\begin{matrix}{{g(N)} = {A^{\prime}{\exp \left( \frac{- E_{a}}{RT} \right)}{\exp \left( \frac{- 1}{f\left( {T,M} \right)} \right)}}} & (15)\end{matrix}$

where f(T,M) is the free volume fraction, or the volume that isunoccupied divided by the total volume, and M is the molecular weight ofthe chain. For this application, chain length N and molecular weight Mcan be used interchangeably (see Equation (29) below) The model in vonMeerwall et al. (1998), see FIG. 6, assumes that each segment has fixedamounts of occupied volume and free volume which do not depend on thechain length, but can depend on temperature and pressure. In addition,the increased mobility of each of the two ends in the chain provideadditional free volume. (By fitting to data, von Meerwall et al. (1998)show that the extra free volume due to an end is roughly the size of thetotal volume of a segment.) By a straightforward calculation, with thispicture, the density and the fractional free volume for mixtures dependon the chain lengths only through the average chain length N.

The function g( N) then depends on the constituents only through theaverage chain length. Note that the molecular average is used, given byEquation (14) while von Meerwall et al. (1999) instead stipulate that itdepends on an average M* defined by

$\begin{matrix}{\frac{1}{M^{*}} = {\sum\limits_{i}\; \frac{v_{i}}{M_{i}}}} & (16)\end{matrix}$

where ν_(i) is the volume fraction of type i (see von Meerwall et al.(1999)). These two averages are not the same, but in many cases givevery similar values, particularly for longer chains.

The data for alkanes in Douglas et al. “Diffusion in ParaffinHydrocarbons,” J. Phys. Chem. 62:1102-1107 (1958) (incorporated byreference herein in its entirety) and for mixtures of alkanes in VanGeet et al., Lo et al. (1998), and Freedman et al. appear to fitEquation (13) quite well as shown in FIG. 7. All data is at 25 to 30° C.and at atmospheric pressure. The solid black line shows the fit for thepure alkanes from C6 to C10 and the binary mixtures of C8 to C12 to apower law dependence on the mean chain length.

In addition, for the live oils in Helbaek et al., shown in FIG. 8, thequantities of products D₀N₀ ^(ν) and D_(g)r_(g) appear to depend only onmean chain length, and even the mixtures with squalene or benzene appearto fit the trend, as shown in FIGS. 7 and 8, respectively. In FIG. 8,the data for live oils from Helbaek et al. is plotted for varioustemperatures and pressures.

The Scaling Laws The free volume theory gives a complicated functionaldependence of D_(i)N_(i) ^(ν) on N. However, as originally observed byMcCall et al. in “Diffusion in Ethylene Polymers IV,” J. Chem. Phys.30(3):771-773 (1959) (incorporated by reference herein in its entirety)for pure alkanes and polymethylene and further verified by the data forpure alkanes in von Meerwall et al. (1998), the diffusion coefficientscan be approximated by a power law over the relevant range of N.According to definition of g(N), this means that g(N) can also beapproximated by a power law in N. Thus, because for mixtures g(N)generally depends on mean chain length, it is expected according toaspects of the invention that it will follow a power law with respect tomean chain length,

g( N )=A N ^(−β)  (17)

where A and β are both slowly varying functions of temperature andpressure. The diffusion coefficient for the i^(th) element is then givenby

D _(i) =AN _(i) ^(−ν) N ^(−β)  (18)

For example, the data for alkanes and mixtures of alkanes (and alsomixtures with squalene) at room temperature and atmospheric pressure isplotted in FIG. 7. A and β depend on temperature and pressure but not onthe constituents. At a temperature equal to 300K and at atmosphericpressure, the line with A=exp(5.6102)×10⁻⁵ cm²/s and β=1.6186 is shownin black.

For live oils, D_(i)r_(i) still depends on the mean chain length.However, as the chain length nears 1, Equation (18) needs to bemodified. The data is still fit well by the function

D _(i) =AN _(i) ^(−ν)( N+1)^(−β)  (19)

Changing N_(i) to N_(i)+1 will not significantly affect the data withthe longer alkanes; however, it will slightly modify the values of A andβ. The data for live oils is plotted in FIG. 8 as a function of N+1. Thefit to Equation (19) for the binary mixtures is shown by the black linein FIG. 8.Equations 18 and 19 are the scaling laws for the diffusion coefficientthat are the basis for this invention. As noted above, scaling lawsgenerally refer to power laws, where one quantity, x, is equal toanother quantity, y, raised to some power ν, so that x=ay^(ν). Althoughν is sometimes determined a priori, often ν and the constant ofproportionality, a, must be determined for the specific properties andconditions of interest.One aspect of this invention is showing that the diffusion coefficientsfollow power laws in both the mean chain length of the mixture and thechain length of the particular alkane, as in Equations 18 and 19.

A second aspect of this invention is to give a method for determiningthe two exponents, ν and β, and the coefficient of proportionality, A,which may depend on temperature and pressure. At a particular pressureand temperature, A and β can be determined by several measurements onknown fluids or liquids. It is noted that there are different methodsfor measuring diffusion coefficients including NMR measurements. Becausethe function g( N) depends only on the mean chain length, it does notmatter which particular alkanes or mixtures are used. In this way, g( N)may be obtained for the values of temperature and pressure of interest.For example, in FIG. 7, the data for pure alkanes and mixtures of C8 andC12 are fit for A and β. They are the intercept and slope of the linesin FIG. 7, respectively. As can be seen in FIG. 7, the data for mixturesof C6 and C16 and mixtures of C6 and C30 also fit this line very well.In later sections, it will be shown how to obtain formulae for thetemperature and pressure dependence of A and β.

Another aspect of this invention is that the scaling relation betweenchain length and diffusion coefficients in Equations (18) and (19) canbe used for fluid typing. Typically, in prior art, such as for polymers,only the scaling with the chain length of the component is given. As thecomposition of the mixture varies, the constant of proportionality willvary. The models in the prior art do not give a method for determininghow the constant of proportionality depends on the composition of themixture, so the chain length distribution of the mixture cannot bedetermined from the prior art. By including the effect of the othercomponents, via the mean chain length, in the scaling relations ofEquations 18 and 19, it becomes possible to solve for the chain lengthsfrom the diffusion measurements. Using Equations (18) or (19) and (14),the mean chain length can be determined from the measured distributionfunction η(D_(i)) of the diffusion coefficients by

$\begin{matrix}{\overset{\_}{N} = {A^{\frac{1}{\upsilon + \beta}}\left( \frac{\sum\limits_{i}\; {{f\left( D_{i} \right)}*\Delta \; D_{i}}}{\sum\limits_{i}\; {{f\left( D_{i} \right)}D^{1\text{/}\upsilon}*\Delta \; D_{i}}} \right)}^{\frac{\upsilon}{\upsilon + \beta}}} & \left( {20a} \right)\end{matrix}$

Once the mean chain length is determined, the distribution of chainlengths may be determined using

$\begin{matrix}{N_{i} = \left( \frac{A}{D_{i}{\overset{\_}{N}}^{\beta}} \right)^{\frac{1}{\upsilon}}} & \left( {20b} \right)\end{matrix}$

Any practioner in the art can see that Equations (20a) and (20b) can becombined by simple mathematical manipulation into a single equation thatgives N_(i) directly in terms of the diffusion distribution. As will bediscussed later, the relaxation times also follow a power law of theform of Equation 18 so these same methods noted above can be used fordetermining chain length distributions from relaxation times.

For mixtures of alkanes, the regularized inverse Laplace transform ofthe NMR data will give a distribution of diffusion coefficients, whichcan be inverted for the chain length distribution, as described above.One caveat to using the inverse Laplace transform is that it will stillgive a relatively broad distribution, even in the case where there isonly one or two diffusion coefficients, as shown in FIG. 1.

FIGS. 9( a)-(d) show an example where the inversion is applied to twodifferent crude oils, both containing a relatively large amount ofsaturates, over 85%, so the alkane model may be applicable. Thedifference between the narrow and broad distributions is clearlyreflected in the chain length distributions found by applying the theoryfrom the polymer models to the NMR diffusion data.

FIGS. 9( a) and (b) show the diffusion distributions found from NMR dataat T=30° C. FIGS. 9( c) and (d) show the chain length distributions. Thechain length distribution from the gas chromatography is plotted withthe circles (“o”) and the chain length distribution found from thedistribution of diffusion coefficients and scaling laws is plotted withthe heavy black line. Equations (20a) and (20b) were used to calculatethis chain length distribution. The values of A and β are the ones foundin the fit in FIG. 7).

The polymer model can also be used to find the viscosity of a mixture,given the distribution of diffusion coefficients. For a polymer,according to the Rouse and Zimm models, the viscosity is related to therotational diffusion coefficient D_(R) by (see Doi et al. and Ferry)

$\begin{matrix}{\eta = {b^{\prime}\frac{c}{N}\frac{kT}{D_{R}}}} & (21)\end{matrix}$

In this equation, c is the number of segments per unit volume and isrelated to the density ρ by c=ρN/M, where M is the mass of the chain.The constant b′ depends on whether the Rouse or Zimm model is used. Forboth the Rouse and Zimm models without excluded volume effects, therotational and translational diffusion coefficients are related by

$\begin{matrix}{D_{R} \propto \frac{D}{{Nl}^{2}}} & (22)\end{matrix}$

Again, the constant of proportionality depends on whether the Rouse orZimm model is used. Combining equations (21) and (22) gives the relationbetween the viscosity and the translational diffusion coefficient:

η=cl ² bkT/D  (23)

where b is a constant that depends on which model is used. For the Rousemodel it is 1/36 and for the Zimm model it is 0.0833.

Note that the product ηD/T is independent or nearly independent of chainlength (see Lo et al. (2000) and Freedman et al). This would not be thecase for hard spheres, where the product would be expected to scale withthe chain length. Instead, in the polymer models the chain lengthscaling drops out due to the “anamolous” dependence on chain length ofboth diffusion coefficients.

The applicability of these equations for mixtures of short chainmolecules and oils, as opposed to polymers, can be checked morequantitatively by comparing the predictions for the values of ηD/T fromthe polymer models with those found experimentally. For alkanes andrefined oils, ηD/T was found to be 3.90⁻⁹ cpcm²/sK; for alkanes andcrude oils, it was found to be 5.05×10⁻⁸ cpcm²/sK (see Lo et al. (1998)and Freedman et al.). If the density is taken to be ρ≈0.8 g/cm³ and theeffective segment length to be l=√{square root over (6.67)}×1.54 Å,which is appropriate for long chains (see Flory, Statistical Mechanicsof Chain Molecules, John Wiley & Sons, Inc., New York (1969),incorporated by reference herein in its entirety), then the Rouse modelgives Dη/T=2.1×10⁻⁸ cpcm²/sK. For the lighter alkanes, the Zimm modelshould be more appropriate. For chain lengths around 10, the effectivedistance between segments is better given by (Flory) l≈√{square rootover (4)}×1.54 Å and the density is closer to ρ√0.75 g/cm³, in whichcase the Zimm model gives Dη/T=3.6×10⁻⁸ cpcm²/sK. A fit to the purealkanes gives 3.8×10⁻⁸ which is well within acceptable limits (see FIG.10).

In a mixture, according to the polymer models (see Ferry), the viscosityis just a sum of the viscosity of each component in the mixture,weighted by the number of molecules of that component per unit volume.Thus, the total viscosity is

$\begin{matrix}{\eta = {\sum\limits_{i}{\frac{{\# \mspace{14mu} {of}\mspace{14mu} i^{th}\mspace{11mu} {molecule}}\;}{{unit}\mspace{14mu} {volume}}\frac{kT}{\left( D_{R} \right)_{i}}}}} & (24)\end{matrix}$

The relation between the translational and rotational diffusioncoefficients then gives

$\begin{matrix}{\eta = {{bckT}{\sum\limits_{i}{y_{i}\text{/}D_{i}}}}} & (25)\end{matrix}$

where y_(i) is the weight fraction of the component.

In FIG. 10, the measured viscosity of mixtures of alkanes is plottedversus the calculated viscosity. Some of the data in this graph isobtained from Rastorguyev et al., Fluid Mechanics—Soviet Research,volume 3, page 156 (1974) (incorporated by reference herein in itsentirety). They agree remarkably well for such a simple model with noadjustable parameters. Some deviation at the high and low ends of theviscosity range is expected because they are nearing their boiling orfreezing point.

Accounting for the Effects of Pressure

It may be preferred that the above model be refined to account for theeffects of pressure and temperature on NMR measurements, such asdiffusion coefficient and relaxation time. As will be shown below,diffusion coefficient D_(i) and relaxation times T_(1,2i) are functionsof density, instead of depending on both pressure and mean chain lengthindependently. D_(i) and T_(1,2i) can be determined at elevatedpressures if the composition of the oil is known.

As discussed above, the free volume fraction depends on the volume/endν_(e), the free volume/segment ν_(sf), the occupied volume/segmentν_(so), and the mean chain length N. For the purposes of thisdiscussion, it is assumed that the occupied volume does not depend onpressure; only the free volume does. However, for consistency, the sameassumption will not be used with respect to temperature dependence(discussed below). This is at least partially justified becausetemperature changes the intrinsic motion of the molecule, which canaffect how much volume it effectively occupies. Once the pressurebecomes too high, these assumptions cannot be expected to hold becausehigh pressure may affect the configuration of the molecules, which canin turn affect the occupied volume.

The free volume fraction can be written in terms of the mean chainlength and various volumes as follows:

$\begin{matrix}{f = \frac{{\overset{\_}{N}v_{sf}} + {2v_{e}}}{{\overset{\_}{N}\left( {v_{so} + v_{sf}} \right)} + {2v_{e}}}} & (26)\end{matrix}$

Density may also be written in terms of these parameters. For a purefluid, the density is given by:

$\begin{matrix}{\rho = \frac{M}{v_{T}}} & (27)\end{matrix}$

where ν_(T) is the total volume/mole and M=14.016N+2.016 is themolecular mass in grams/mole. For a mixture, the total volume is givenby:

ν_(T) = Nν _(s)+2ν_(e)  (28)

the number of grams/mole is given by:

M=14.016 N+2.016  (29)

and the expression for the density becomes

$\begin{matrix}{\rho = \frac{{14.016\overset{\_}{N}} + 2.016}{{\overset{\_}{N}v_{s}} + {2v_{e}}}} & (30)\end{matrix}$

Next, in both the expression for the free volume fraction and thedensity, the pressure and N-dependent parts may be separated from therest as follows:

$\begin{matrix}{{f = \frac{\left\lbrack {{v_{sf}(P)} + {2{{v_{e}(P)}/\overset{\_}{N}}}} \right\rbrack}{v_{so} + \left\lbrack {{v_{sf}(P)} + {2{{v_{e}(P)}/\overset{\_}{N}}}} \right\rbrack}}{and}} & (31) \\{\rho = {\frac{14.016 + {2.106/\overset{\_}{N}}}{v_{so} + {v_{sf}(P)} + {2{{v_{e}(P)}/\overset{\_}{N}}}} \approx \frac{14}{v_{so} + {v_{sf}(P)} + {2{{v_{e}(P)}/\overset{\_}{N}}}}}} & (32)\end{matrix}$

Thus, in both the density and the free volume fraction, the onlydependence on the pressure and mean chain length is through thecombination

h( N,P)=ν_(sf)(P)+2ν_(e)(P)/ N   (33)

In other words, the density and free volume fraction may be written as:

$\begin{matrix}{\rho = \frac{14}{v_{so} + {h\left( {\overset{\_}{N},P} \right)}}} & \left( {34a} \right) \\{f = \frac{h\left( {\overset{\_}{N},P} \right)}{v_{so} + {h\left( {\overset{\_}{N},P} \right)}}} & \left( {34b} \right)\end{matrix}$

Then the free volume fraction can be written in terms of the density asfollows:

$\begin{matrix}{f = \frac{14 - {v_{so}\rho}}{14}} & (35)\end{matrix}$

With the assumption that ν_(so) is independent of pressure and chainlength, all the pressure and chain length dependence of f can instead bereplaced by the dependence off on density. According to Equations (35)and (13) and (15) then, at a fixed temperature, the rescaled diffusioncoefficient D_(i)N_(i) ^(ν) and relaxation times T_(1i)N_(1i) ^(k) andT_(2i)N_(2i) ^(k) should depend only on density, regardless of the meanchain length and pressure. In other words, according to an aspect of theinvention, the rescaled diffusion coefficients and relaxation timesshould be functions only of density and temperature.

In FIG. 11, T_(l) data for C8, C10, C12, and C16 from Zega's PhD thesis,“Spin Lattice Relaxation in Normal Alkanes at Elevated Pressures”, RiceUniversity (1991) (incorporated by reference herein in its entirety) isplotted as a function of density. The temperature ranges from 25° C. to85° C. For C8 through C12, the pressure ranges from about 20 psia to 600psia (41 MPa), while for C16, the pressure only goes up to about 400psia (28 MPa). In this plot, there is no clear connection between therelaxation times of each of the alkanes, except that it increases as thechain gets shorter.

FIG. 12 shows the scaled relaxation time N^(k)T_(l)(k=1.24) as afunction of density. As can be seen in this figure, the data nowcollapse to four lines, one for each of the four temperatures. The mainexception is for octane at a high temperature, and, to a lesser degree,the hexadecane at low temperatures. Presumably, these discrepanciesoccur because the octane is getting too close to its boiling point andthe hexadecane is getting too close to its melting point, but therecould be other explanations for why the range of validity of thecollapse is limited. Apart from these limiting cases, the collapse isquite remarkable and demonstrates that for a range of pressures andchain lengths, the scaled relaxation time depends only on temperatureand density.

FIG. 13 shows the data at 25° C. from (1) Zega's PhD thesis, (2) Zega'sMaster's thesis, “Spin Lattice Relaxation in Pure and Mixed Alkanes andTheir Correlation with Thermodynamic and Macroscopic TransportProperties”, Rice University (1988) (incorporated by reference herein inits entirety), and (3) Zega, et al., “A Corresponding-States Correlationof Spin Relaxation in Normal Alkanes,” Physics A, Volume 156, pages277-293 (1989) (incorporated by reference herein in its entirety). Thisfigure includes data for pure alkanes from C6 to C16 and data formixtures of C6 and C16 at atmospheric pressure as well as the data atelevated pressure. Again the data collapses to a line, showing that itprimarily depends only on density. Also, the agreement between the datafor the mixtures and pure alkanes at atmospheric pressure demonstratesthat the scaling law within a mixture T_(1i)∝N_(i) ^(−k), is valid, andthat the product T_(1i)N_(i) ^(−k) depends only on the mean chainlength. The small systematic discrepancies can be due to the fact thatas the chains get closer to hexadecane, they are also getting closer totheir melting point. In addition, the scaling law does not reallyaccount for intermolecular relaxation, which can vary as the chainlengths vary.

Next, the dependence of the diffusion coefficient on density isanalyzed. In FIG. 14, the scaled diffusion coefficient D_(i)N_(i) ^(ν)for pure alkanes and some binary mixtures at 25° C. is plotted as afunction of density. The data for C6 and C8 from Harris's “Temperatureand Density Dependence of Self-Diffusion of n-hexane from 223 to 333 Kand up to 400 MPa”, J. Chem. Soc., Faraday Trans. Volume 1, Issue 78,pages 2265-2274 (1982) (incorporated by reference herein in itsentirety) and Harris et al.'s “Temperature and Density Dependence ofSelf-Diffusion Coefficients of Liquid n-octane and toluene,” Mol. Phys.Volume 78, Issue 1, pages 235-248 (1993) (incorporated by referenceherein in its entirety) range from atmospheric pressure to about 350MPa. The data for C16 is from Dymond et al.'s “The Temperature andDensity Dependence of the Self-Diffusion Coefficient of n-hexadecane,”Mol. Phys., Volume 75, Issue 2, pages 461-466 (1992) (incorporated byreference herein in its entirety) ranges from atmospheric pressure to 27MPa. All the other data points are at atmospheric pressure. The equationfor density from von Meerwall et al. (1998) was used to obtain thedensity at atmospheric pressure.

The data collapse reasonably well to a single line. In fact, over theentire range, the agreement between hexane and octane is quiteremarkable. However, as the density nears 0.75 g/cm³, the scaleddiffusion coefficients for hexane and octane at elevated pressure startto deviate noticeably from the scaled diffusion coefficient for dodecaneat atmospheric pressure, and the difference between them and C16 appearsto be significant. This deviation occurs at about 250 MPa for C6 and 100MPa for C8. At these rather high pressures the assumptions about freevolume may no longer be valid. In addition, at these high pressures, theequations for the density of hexane and octane may not be valid.

Accordingly, the free volume and, hence, the diffusion coefficients andrelaxation times are functions of the density. Accordingly, if thescaling laws are known at the temperature of interest and at onereference pressure, then the relationship between diffusion coefficientsand composition can be determined at any pressure, as long as thedensity of the oil is known. This means that chain length distributioncan be determined from a measurement of both density and the diffusionor relaxation distribution.

For the following discussion, it is assumed that the equation for thediffusion coefficient at atmospheric pressure P₀ and T is known. If asample is measured at another pressure P and has density P at thispressure, an effective chain length N_(eff) can be defined for thesample as follows. The effective chain length is the chain length thathas the same density as the sample, at atmospheric pressure. Thus,

ρ(N _(eff) ,T,P ₀)=ρ( N,T,P)=ρ  (36)

Using Equation (30) for density,

$\begin{matrix}{\rho = \frac{M}{{N_{eff}v_{s}} + {2v_{e}}}} & (37)\end{matrix}$

where M=14.016N_(eff)+2.016, and ν_(s) and ν_(e) are given by theirvalues at atmospheric pressure. The effective chain length is then givenby

$\begin{matrix}{N_{eff} = \frac{{2v_{e}\rho} - 2.016}{14.016 - {v_{s}\rho}}} & (38)\end{matrix}$

Equations of von Meerwall et al. (1998) may be used to determine thevalues of ν_(s) and ν_(e). In von Meerwall et al. (1998), the density atatmospheric pressure is given by

ρ(T,N,P ₀)=[1/ρ_(∞)(T)+2V _(E)(T)]⁻¹  (39)

where

1/ρ_(∞)(T)=[1.142+0.00076T(° C.)±0.005]cm³/g  (40)

and

V _(E)(T)=[13.93+0.060T(° C.)±0.3]cm³/mol  (41)

Setting Equation (39) equal to Equation (37) and substituting in thevalue for the mass M, ν_(s) and ν_(e) may be solved for. Thevolume/segment is given by

ν_(s)=14.016 gmol⁻¹/ρ_(∞)(T)  (42)

and the extra volume/end is given by

ν_(e)=2.016 gmol⁻¹/ρ_(∞)(T)+2V _(E)  (43)

Next, the diffusion coefficient is calculated at elevated pressure. Attemperature T, the scaled diffusion coefficient is a function ofdensity, so that

N _(i) ^(ν) D _(i)(ρ(N,T,P))=N _(i) ^(ν) D _(i)(ρ(N _(eff) ,T,P₀))  (44)

Over some range of temperatures and chain lengths, the diffusioncoefficient at atmospheric pressure has the form

D _(i)(ρ(N _(eff) ,T,P ₀)=A(T,P ₀)N _(i) ^(−ν) N _(eff) ^(−β(T,P) ⁰⁾  (45)

Above it was shown (see discussion of Equation (18)) thatA(T,P₀)=exp(5.6102)×10⁻⁵ cm²/s and β(T,P₀)=1.6186 at 300 K for chainlengths from about C5 to C16; the value for A(T,P₀) and β(T,P₀) will bedetermined for a wide range of temperatures and chain lengths below inthe discussion relating to temperature effects. Combining Equations (44)and (45), the diffusion coefficient at pressure P is given by

D _(i)(T,P)=A(T,P ₀)N _(i) ^(−ν) N _(eff) ^(−β(T,P) ⁰ ⁾  (46)

where N_(eff) is given by Equations (38) through (43).

A similar calculation for the relaxation times yields

T _(1i)(T,P)=T _(2i)(P)=B(T,P ₀)N _(i) ^(−k) N _(eff) ^(−γ(T,P) ⁰⁾  (47)

where B(T,P₀) and γ(T,P₀) are the values of B and γ at atmosphericpressure. The version of the scaling laws in Equations (46) and (47)along with the original scaling laws (Equations (18) and (48)) are anaspect of the invention and are the basis for the third embodiment ofthe invention. To illustrate these equations, in FIG. 15, T₁ data fromZega's PhD thesis (see above) is plotted as a function of N_(eff). Foreach value of T, a separate power law is obtained. The fit for the powerlaws are shown by the solid lines. At 25° C., a separate fit wasperformed for the data at elevated pressures A and the data atatmospheric pressure B. The fit at atmospheric pressure demonstrates thescaling law in mean chain length and thus directly validates theoriginal scaling law for T₁ given by

T _(1i) =B(T,P)N _(i) ^(−k)( N )^(−γ(T,P))  (48)

This equation is an aspect of the invention. It is the scaling relationfor the NMR relaxation times used in all embodiments of the invention.Because it has the same form as the scaling relation for the diffusioncoefficients Equation (18), it can be used to determine chain lengthdistributions from relaxation measurements.

For live oils, the reference pressure is preferably not equivalent tothe atmospheric pressure for the following reasons: (1) the scaling lawis extrapolated beyond the range where it was fit; (2) the scaling lawis applied to a regime of short chains which is not truly physicalbecause at atmospheric pressure the alkanes are no longer liquids forchain lengths less than C5; and (3) the linear relationship is used formolar volume versus N_(eff)(P₀) for the short chains, when this relationdoes not hold. In addition, for the longer chains, getting a ‘fit’ isambiguous because it depends on the pressure range of interest.

Instead, the scaling law for live oils should be determined using aneffective chain length that is defined at an elevated pressure P, wherethe full range of N_(eff)(P) is covered by the calibration data andwhere the oil is a liquid or supercritical for the full range ofN_(eff). To go from the scaling law in N_(eff)(P₀)+1 where P₀ is theatmospheric pressure to the plot as a function of N_(eff)(P₀)+1 where P₀is an elevated pressure, the density data from the NIST webbook can beused to find the density of all the alkanes up to C7. The data at C6 andC7 is used to find the molar volume as a function of chain length (asshown in FIGS. 16-18). The slope and intercept (ν_(s) and 2ν_(e),respectively) are determined using Equation (38). Then the density ofthe oil sample is substituted into Equation (38) to obtain the effectivechain length to obtain N_(eff)(P₀) where P₀ is now the elevatedpressure.

This method works when N_(eff) is in the physical range of chain lengthsor densities and density is not so high or so low that other physicscome into play. This method also works primarily when the fluid sampleis a mixture of only alkanes because it is very sensitive to theaddition of other molecules while D and T_(1,2) are not necessarilyaffected. Thus, if elevated-pressure mixtures having other molecules inaddition to alkanes (such as aromatics and asphaltenes, etc.) are underconsideration, a more robust way is needed to take into account pressureeffects. The ideas above regarding N_(eff) and free volumes may beapplied to a method to calculate the pressure effects that does notrequire measuring the density of the oil sample. Instead, it onlyrequires knowing the density of alkanes at the pressures andtemperatures of interest (plus a reference pressure), which can be foundin standard tables such as the NIST webbook. This method is described inmore detail below.

First, the pressure dependence of the molar volume will be examined. Themolar volume is given by

ν_(T)=ν_(s) N+2ν_(e)  (49)

where ν_(e) is the free volume per end and the volume per segment ν_(s)can be broken into the occupied volume per segment ν_(so) and the freevolume per segment ν_(sf). In Kurtz, Jr., “Physical Properties andHydrocarbon Structure,” Chemistry of Petroleum Hydrocarbons (Brooks, etal. editors) pages 275-331 (1954) (incorporated by reference herein inits entirety), it was found that ν_(e) varies much more strongly withpressure than ν_(s) does. However, at very high pressures, thisdifference decreases significantly. The molar volumes ν_(e) and ν_(s)can be determined by looking at density data at a fixed temperature andpressure and plotting the volume ν_(T)=M/p as a function of chainlength. Several examples are provided in FIGS. 16 through 18( f).

In FIG. 16, the molar volumes calculated from the densities from Zega'sPhD Thesis (see above) are plotted as a function of chain length at twodifferent temperatures and three different pressures. As can be seen inthe plot, for each temperature and pressure, the data points lie on astraight line. As the temperature and pressure are changed, theintercept changes more than the slope, although the slope does change alittle as the temperature is raised.

The values for the slope ν_(s) and intercept 2ν_(e) of the fitted lines(shown by the solid lines) are given in Table 1. For comparison, theslope and intercept calculated from Equations (42) and (43) atatmospheric pressure (0.1 MPa) are also given in the table, as well asvalues at 50 MPa, which are calculated from the data of Lemmon et al.,“Thermophysical Properties of Fluid Systems,” NIST Chemistry WebBook,NIST Standard Reference Database Number 69 (Linstrom et al. editors),March 2003 (http://webbook.nist.gov) (incorporated by reference hereinin its entirety) as in the examples below.

Intercept 2v_(e) Temperature (° C.) Pressure (MPa) Slope v_(s) (cm³/mol)(cm³/mol) 25° 0.1 16.27 33.20 0.14 16.32 32.61 20.7 16.18 30.00 41.416.01 28.45 50.0 16.07 27.06 85° 0.1 16.91 40.49 0.14 16.84 40.84 20.716.74 35.56 41.4 16.54 33.01

FIG. 17 is an example of both live and dead alkanes at T=50° C. For N>3,the fit to a straight line is adequate, and as the pressure is raised,the fit improves for smaller chain lengths. The fits to C6 and C7extrapolate well to the molar volume for C16. The slope ν_(s) andintercepts 2ν_(e) of these lines are provided in Table 2 (below). Thedata for C1 through C7 is from the NIST webbook, while the data for C16is from Dymond et al. (see above).

Intercept 2v_(e) Temperature (° C.) Pressure (MPa) Slope v_(s) (cm³/mol)(cm³/mol) 30° 0.1 16.33 33.81 30.0 16.08 30.31 40.0 16.09 28.79 50.016.09 27.53 50° 0.1 16.54 36.24 23.4 16.12 34.08 50.9 16.20 29.19 60° C.0.1 16.65 37.45 30.0 16.22 33.72 40.0 16.25 31.74 50.0 16.27 30.12

The densities for live mixtures of C1 with C6 and C2 with C6 are shownin FIGS. 18( a)-(f). As long as N is greater than about 2 or 3, thevolumes for the mixtures are quite close to the interpolated volumes forthe pure substances, and they all lie close to the straight lines. FIGS.18( a)-(f) show the fits to C6 and C7 and the slope and intercept ofthese lines are given in Table 2. The data for the pure alkanes was fromthe NIST webbook, and the data for the mixtures is from Friend, “MSTMixture Property Database,” NIST Standard Reference Database 14, October1992 (incorporated by reference herein in its entirety). In Table 2, thevalues of ν_(s) and 3ν_(e) at atmospheric pressure (0.1 MPa) were foundusing Equations (42) and (43).

In both Table 1 and 2, the slope ν_(s) changes very little as a functionof pressure, while the intercept 2ν_(e) changes more rapidly. The sametrend is true for the temperature dependence. However, both the slopeand intercept appear to depend more strongly on temperature than onpressure. Also, ν_(s) and 2ν_(e) change more with pressure at highertemperatures than at lower temperatures. The fact that ν_(s) changes solittle with pressure supports the assumptions that the occupied volumeper segment ν_(so) does not depend on pressure.

In order to determine the pressure dependence on the diffusioncoefficients, the effective chain length N_(eff) is revisited. A fluidwith effective chain length N_(eff) has the same density at thereference pressure P₀ as the sample has at its pressure P. For thefollowing discussion, P₀ will no longer be restricted to atmosphericpressure. N_(eff) can be written in terms of the mean chain length ofthe sample as follows, by definition of N_(eff),

ρ(N _(eff) ,P ₀)=ρ( N,P)  (50)

The expression for ρ in Equation (32) is now used to solve for N_(eff):

$\begin{matrix}{N_{eff} = {\frac{2{v_{e}\left( P_{0} \right)}}{{\overset{\_}{N}\left( {{v_{s}(P)} - {v_{s}\left( P_{0} \right)}} \right)} + {2{v_{e}(P)}}}\overset{\_}{N}}} & (51)\end{matrix}$

Based on the values of ν_(s) and 2ν_(e) of Tables 1 and 2, the change inν_(s) with pressure is much smaller than the value of 2ν_(e), so, unlessN gets large (i.e., N(ν_(s)(P)−ν_(s)(P₀)) is not much smaller than2ν_(e)(P)), the first term of the denominator can be dropped to obtain:

$\begin{matrix}{N_{eff} = \frac{{v_{e}\left( P_{0} \right)}\overset{\_}{N}}{v_{e}(P)}} & (52)\end{matrix}$

Now the diffusion coefficient at pressure P may be determined. Asbefore,

D _(i)(ρ( N,P))=D _(i)(ρ(N _(eff) ,P ₀))  (53)

From the scaling law for D(N_(eff),P₀),

D _(i)(ρ( N,P)=A(T,P ₀)N _(i) ^(−ν) N _(eff) ^(−β(T,P) ⁰ ⁾  (54)

Substituting in the expression for N_(eff).

$\begin{matrix}{{D_{i}\left( {\rho \left( {\overset{\_}{N},P} \right)} \right)} = {{A\left( {T,P_{0}} \right)}\left( \frac{v_{e}\left( P_{0} \right)}{v_{e}(P)} \right)^{- {\beta {({T,P_{0}})}}}N_{i}^{- \upsilon}{\overset{\_}{N}}^{- {\beta {({T,P_{0}})}}}}} & (55)\end{matrix}$

In this way, the scaling law as a function of N for D_(i) is obtained.The exponent β is independent of pressure, so the scaling law has theform

$\begin{matrix}{{D_{i}\left( {\rho \left( {\overset{\_}{N},P} \right)} \right)} = {{A\left( {T,P_{0}} \right)}\left( \frac{v_{e}\left( P_{0} \right)}{v_{e}(P)} \right)^{- {\beta {(T)}}}N_{i}^{- v}{\overset{\_}{N}}^{- {\beta {(T)}}}}} & (56)\end{matrix}$

The pressure dependence for the relaxation times can be found in asimilar way, and has the form

$\begin{matrix}{{T_{1,{2i}}\left( {\rho \left( {\overset{\_}{N},P} \right)} \right)} = {{B\left( {T,P_{0}} \right)}\left( \frac{v_{e}\left( P_{0} \right)}{v_{e}(P)} \right)^{- {\gamma {(T)}}}N_{i}^{- k}{\overset{\_}{N}}^{- {\gamma {(T)}}}}} & (57)\end{matrix}$

Equations (56) and (57) are an aspect of the invention. They give thepressure dependence of the constants of proportionality, A and B, andthe exponents, β and g, which are used in the fifth and sixthembodiments of the invention.

If information about the density of alkanes at the desired temperature Tand at both the desired pressure P and the reference pressure P₀ isknown, then ν_(e)(P₀) and ν_(e)(P), which both depend on T, can be fit.For example, the data in the NIST webbook can be fit to C6 and C7 as inFIGS. 17 and 18( a)-(f). Then, as long as β(T) and A(P₀,T) are known,the diffusion coefficient (and similarly the relaxation times) can bedetermined at any pressure.

An example is provided in FIG. 19, which shows the scaling law atatmospheric pressure and 25° C. The values of ν_(e)(P₀) and ν_(e)(P)with P₀=0.1 MPa and P=50 MPa are taken from Table 2. The fit to Equation(56) is shown, where the parameters A(P₀,T) and β(T) are the values fromthe scaling law at atmospheric pressure.

Thus, once the scaling law at atmospheric pressure is known as well assome densities at atmospheric pressure and at 50 MPa, a reasonably goodfit to the diffusion coefficients at 50 MPa may be obtained with nofitting parameters.

One of the consequences of the pressure-independence of β(T) is that, ona log plot, the curves for diffusion coefficients as a function oftemperature or plots of distributions of diffusion coefficients will alllie parallel to each other as the pressure is changed. However, once thepressure becomes very high, the parameter β(T) will have some pressuredependence, as found by Vardag et al., for pressure above about 100 MPa(above about 100 MPa, the slope starts to change). This presumably is anindication that the occupied volume per segment is also affected bypressure at these high pressures. As described below, it is probablyalso an indication that the diffusion coefficient depends on pressure aswell as density at these higher pressures. (This, in turn, shouldindicate that at high pressures the occupied volume depends onpressure.)

The fact that the exponent β(T) is independent of pressure (at pressuresthat are not extremely high) follows from the fact that the free volumefrom the ends is much larger than the change in volume of the segmentsas the pressure is changed. It also turns out that it is a consequenceof requiring the diffusion coefficient D both to depend on pressure andchain length only through density and to follow scaling laws at anypressure. For example, if Equation (51) is substituted into Equation(54) for the effective chain length N_(eff) without imposing anycondition on the sizes of ν_(s) and ν_(e),

$\begin{matrix}{{D_{i}\left( {T,P} \right)} = {{A\left( {P_{0},T} \right)}{N_{i}^{- v}\left\lbrack \frac{2{v_{e}\left( P_{0} \right)}}{{\overset{\_}{N}\left( {{v_{s}(P)} - {v_{s}\left( P_{0} \right)}} \right)} + {2{v_{e}(P)}}} \right\rbrack}^{- {\beta {({T,P_{0}})}}}{\overset{\_}{N}}^{- {\beta {({T,P_{0}})}}}}} & (58)\end{matrix}$

If D( N) is required to obey a strict scaling law in N, the relation inEquation (58) is only possible if the expression in the denominator isreplaced with 2ν_(e)(P). Thus, the scaling law is only possible ifN(ν_(s)(P)−ν_(s)(P₀))<<2ν_(e)(P). In that case, the equation of D,reduces to the scaling law of Equation (56).

For short chains, the molar volume is no longer a linear function ofchain length. At the temperatures of FIGS. 21 and 22( a)-(f), thisoccurs for chain lengths less than or equal to about 2, but as thetemperature is raised, it occurs at longer chain lengths. In addition,the scaling law is no longer in N, but in N+1. Thus, at pressure P, thediffusion coefficient can be written in terms of the effective chainlength at the reference pressure P₀ as follows

D _(i)(ρ( N,P))=A(T,P ₀)r _(i)(N _(eff)(p ₀ ₎+1)^(−β(T,P) ⁰ ⁾  (59)

where r_(i)=N_(i) ^(ν) for the alkanes with chain length larger thanabout 5, and r_(i) is proportional to an effective hard sphere radiusfor smaller chain lengths. If the value of N_(eff) from Equation (52) issubstituted into the scaling law for live oils, still assuming thatN(ν_(s)(P)−ν_(s)(P₀))<<2ν_(e)(P), then

$\begin{matrix}{{D_{i}\left( {\rho \left( {\overset{\_}{N},P} \right)} \right)} = {{A\left( {T,P_{0}} \right)}\left( \frac{v_{e}\left( P_{0} \right)}{v_{e}(P)} \right)^{- {\beta {(T)}}}{r_{i}\left( {\overset{\_}{N} + \frac{v_{e}\left( P_{0} \right)}{v_{e}(P)}} \right)}^{- {\beta {(T)}}}}} & (60)\end{matrix}$

In all the examples above, the extra free energy from the edges ν_(e)has not changed by more than about 20% as the pressure was varied. It isvery useful to have a scaling law in N+1 for the live oils. Thus, theexpression ν_(e)(P)/ν_(e)(P₀) will be replaced with 1 in Equation (60)to obtain a useful expression for the pressure dependence of live oils.

$\begin{matrix}{{D_{i}\left( {\rho \left( {\overset{\_}{N},P} \right)} \right)} \approx {{A\left( {T,P_{0}} \right)}\left( \frac{v_{e}\left( P_{0} \right)}{v_{e}(P)} \right)^{- {\beta {(T)}}}{r_{i}\left( {\overset{\_}{N} + 1} \right)}^{- {\beta {(T)}}}}} & (61)\end{matrix}$

The fit to a scaling law does not seem to be that sensitive to thisapproximation, as will be described below. Equation (61) is amodification of Equation (56), which is used in the fifth and sixthembodiments of the invention when the mean oil is expected to have asignificant amount of dissolved gas so the mean chain length is close toone (i.e. roughly less than 3).

FIG. 20 shows that the data at different pressures can be collapsed to asingle line by using the pressure dependence of Equation (61). In thisfigure, data from Helbaek is plotted as a function of the mean chainlength. The data is at 30 MPa, 40 MPa, and 50 MPa. Data from Freedman etal. is also included at atmospheric pressure (0.1 MPa). Instead ofplotting, the usual scaled diffusion coefficient, an additional scalefactor of the form (ν_(e)(P₀)/ν_(s)(P))^(β)(T,P₀) is included. Thus,

$\begin{matrix}{{\left( \frac{v_{e}\left( P_{0} \right)}{v_{e}(P)} \right)^{\beta {(T)}}r_{i}{D_{i}\left( {\rho \left( {\overset{\_}{N},P} \right)} \right)}} = {{A\left( {P_{0},T} \right)}\left( {\overset{\_}{N} + 1} \right)^{- {\beta {(T)}}}}} & (62)\end{matrix}$

is plotted versus the actual mean chain length of the alkane or mixture.The reference pressure P₀ is taken to be atmospheric pressure but thecollapse looks very similar if one of the elevated pressures is usedinstead. The value for β(T) used is described below and the values forν_(e)(P₀) and ν_(e)(P) are given in Table 2 and can be found using thedensity data from the NIST webbook as described above.

Note that the quantity on the right hand side of Equation (62) dependsonly on the reference pressure. Thus, the data points at all fourpressures should collapse to a single line. As can be seen in FIG. 20,the data collapses very well to a single line, the scatter in the dataat a single pressure is much larger than the scatter between data atdifferent pressures. The solid lines show the fits to the scaling law inEquation (19) for the three elevated pressures, which again collapse toa single line. The slope of the line for the lowest pressure (30 MPa)does deviate a little from the two lines at high pressure, and a closeinspection of the data points reveals a small systematic tendency forthe data at the lowest pressure to have a slightly different slope.

Accordingly, if the scaling law is known at some temperature andreference pressure and if the dependence on chain length of the molarvolume is known at the reference pressure and any pressure P, then thediffusion coefficients and the relaxation items may be calculated atpressure P. Having determined the pressure dependence, now thetemperature dependence is considered.

Accounting for the Effects of Temperature

The power law dependence on chain length is combined with the Ahrreniustemperature dependence to determine how diffusion and relaxation dependon temperature and chain length. As above, the discussion is focused ondiffusion coefficients, but may be similarly applied to relaxation time.

According to Equation (18), the diffusion coefficient follows a scalinglaw of the form:

D _(i) =A(T,P)N _(i) ^(−ν) N ^(−β() T)  (63)

where A(T,P) and β(T) depend on temperature. Alternatively, for puresubstances, the diffusion coefficient D has been found to have anArrhenius temperature dependence of the form

D∝e^(−E) ^(a) ^((N))/_(kT)  (64)

where the activation energy E_(a)(N) is a function of chain length (seeVardag et al., von Meerwall et al. (1998), Douglass et al., and Ertl etal., “Self-Diffusion and Viscosity of Some Liquids as a Function ofTemperature,” AIChE Journal, Volume 19, Issue 6, pages 19-40 (1973)(incorporated by reference herein in its entirety)). These twoexpressions for the diffusion coefficient are consistent if theactivation energy is logarithmic in N, of the form

E _(a)(N)=b+d log(N)  (65)

for some temperature-independent coefficients b and d. This was in factfound in Ertl et al. and von Meerwall et al. (1998). The diffusioncoefficient can then be written in terms of four temperature-independentparameters a, b, c, and d in the form

D _(i) =e ^(−(a+b/T)) N _(i) ^(−ν) N ^(−(c+d/T))  (66)

In other words, the exponent β(T) in the scaling law is given by

β(T)=c+d/T  (67)

and the coefficient A(T,P) is given by

A(T,P)=exp[a(P)+b(P)/T]  (68)

Because A depends on pressure, the parameters a and b can also depend onpressure. However, since β is independent of pressure, c and d shouldalso be independent of pressure. The temperature dependence for therelaxation times can be found in a similar manner, with the result

T _(1i) =T _(2i) =e ^(−(a′(P)+b′(P)/T)) N _(i) ^(−k) N^(−(c′+d′/T))  (69)

where a′(P), b′(P), c′, and d′ are temperature-independent parameters.Equations (66) through (69) give the temperature dependence of thecoefficients of proportionality, A and B, and the exponents β and γ,which are used in the second, fourth and sixth embodiments of theinvention.

To illustrate this temperature dependence, data for diffusioncoefficients for pure alkanes taken at a wide range of temperatures andat atmospheric pressure (or saturation vapor pressure) are shown. Thediffusion coefficients as a function of reciprocal temperature areplotted in FIG. 21. A four parameter fit was performed for Equations(66) for the diffusion coefficient. All of the data shown in FIG. 26,apart from the data for C8 from Harris et al. and the data for C 78 andC1 54, which are blends of alkanes with mean chain length 78 and 154,respectively, were used in the fit. As can be seen, the data fits quitewell to the Arrhenius plots and the lines match up well with theappropriate chain lengths.

The values of the four fitted parameters were

-   -   a=−6.3326    -   b=143.6869    -   c=−0.2442    -   d=588.4961        when the coefficient A(T,P) is given in 10⁻⁵ cm²/s. The exponent        β(T) varies a fair amount with temperature. At 25° C. it is        β(25° C.)=1.73. At 100° C. it is β(100° C.)=1.33, and at 200° C.        it is β(200° C.)=1.00. This would appear to be an indication        that the occupied volume/segment does depend on temperature. For        these temperatures, the coefficient A(T,P) does not vary by as        great a percentage, and, in fact, a+b/T is not very sensitive to        temperature. For T=25° C., 100° C., 200° C., A(T,P)=347.5×10⁻⁵        cm²/s, 382.8×10⁻⁵ cm²/s, and 415.3×10⁻⁵ cm²/s, respectively.

In order to look at live oils, the data is also fit to Equation (66),with the result

-   -   a=−5.7256    -   b=−212.9887    -   c=−0.4636    -   d=705.1817

With these parameters, the fit to the data looks almost identical to thefit with the parameters for the first scaling law. However, asdetermined above, even though this equation fits the data for the deadoils quite well, it still does not extrapolate well to smaller chainlengths for the live oils.

Next, the T₁ data from Zega's PhD thesis is considered. To demonstratethat the data follows an Arrhenius law, the data for C8, C10, C12, andC16 can be plotted at atmospheric pressure versus the reciprocaltemperature From the slope and intercepts of the resulting lines, it ispossible to find the values of a′, b′, c′, and d′. Instead in FIG. 22,the results of applying a four parameter fit directly to the scaling lawin N_(eff) is shown. Accordingly, the parameters have the values

-   -   a′=−5.75    -   b′=−227    -   c′=−1.43    -   d′=755        when T₁ is in units of seconds. As with the equation for the        diffusion coefficients, these parameters can be used to        interpolate for relaxation times at different chain lengths and        temperatures. When combined with density data, it can also be        used to find the relaxation times at different pressures.

It is noted that while the above applications relate to oilapplications, the method may be adapted for other applications includingthe medical and food preparation industries, for example.

While the invention has been described herein with reference to certainexamples and embodiments, it will be evident that various modificationsand changes may be made to the embodiments described above withoutdeparting from the scope and spirit of the invention as set forth in theclaims.

1. A method for determining the characteristics of a fluid sample,comprising: a. obtaining measurements of a plurality of calibrationsamples at a first temperature and a reference pressure, wherein thecalibration samples have differing mean chain lengths, and wherein saidmeasurements are selected from the group consisting of diffusionmeasurements and relaxation measurements; b. determining the density ofmore than one pure alkane or mixtures of alkanes at the firsttemperature and the reference pressure, wherein density is determined asa function of mean chain length; c. obtaining measurements of a sampleunder investigation at the first temperature and a second pressure,wherein said measurements are selected from the group consisting ofdiffusion measurements and relaxation measurements; d. determining thedensity of the sample under investigation at the first temperature andthe second pressure; e. applying the density function of (c) to thedensity measurements of (d) and using the measurements of (a) todetermine the scaling law at the second pressure in terms of chainlength; f. applying the scaling law of (e) to the data of (c) todetermine the composition of the sample under investigation.
 2. Themethod of claim 1, wherein said density measurements of (b) are obtainedfrom standard look-up tables.
 3. The method of claim 1, wherein (f)further comprises determining the distribution of chain lengths of theconstituents of said sample under investigation.
 4. The method of claim1, wherein the density measurement of (d) is selected from the groupconsisting of mass density, carbon density, and hydrogen index.
 5. Amethod for determining the characteristics of a fluid sample,comprising: a. obtaining measurements of a plurality of calibrationsamples at a reference pressure and at more than one temperature,wherein the calibration samples have differing mean chain lengths, andwherein said measurements are selected from the group consisting ofdiffusion measurements and relaxation measurements; b. determining thedensity of more than one pure alkane or mixtures of alkanes at thereference pressure and at a temperature within or near the range oftemperatures in (a), wherein density is determined as a function of meanchain length; c. obtaining measurements of a sample under investigationat a second pressure and at a temperature within or near the range oftemperatures in (a), wherein said measurements are selected from thegroup consisting of diffusion measurements and relaxation measurements;d. determining the density of the sample under investigation at thesecond pressure and at a temperature within or near the range oftemperatures in (a); e. applying the density function of (c) to thedensity measurements of (d) and using the measurements of (a) todetermine the scaling law at the second pressure in terms of chainlength; f. applying the scaling law of (e) to the data of (c) todetermine the composition of the sample under investigation.
 6. Themethod of claim 5, wherein said density measurements of (b) are obtainedfrom standard look-up tables.
 7. The method of claim 5, wherein (f)further comprises determining the distribution of chain lengths of theconstituents of said sample under investigation.
 8. The method of claim5, wherein the density measurement of (d) is selected from the groupconsisting of mass density, carbon density, and hydrogen index.
 9. Amethod for determining the characteristics of a fluid sample,comprising: a. obtaining measurements of a plurality of calibrationsamples at a first temperature and a reference pressure, wherein thecalibration samples have differing mean chain lengths, and wherein saidmeasurements are selected from the group consisting of diffusionmeasurements and relaxation measurements; b. obtaining measurements ofsaid sample under investigation at the first temperature and a secondpressure, wherein said measurements are selected from the groupconsisting of diffusion measurements and relaxation measurements; c.determining the relationship of volume of one or more alkanes ormixtures of alkanes to (i) the mean chain length at the firsttemperature and the reference pressure and (ii) the mean chain length atthe first temperature and a second pressure; d. determining the scalinglaw as a function of chain length, using the functions of (c) and themeasurements of (a); e. applying the scaling law of (d) to themeasurements of (b) to determine the composition of the sample underinvestigation.
 10. The method of claim 9, wherein the scaling law of (d)is a function of chain length and mean chain length.
 11. The method ofclaim 9, wherein said volume of (c) are obtained using standard look-uptables.
 12. The method of claim 9, wherein the volumes of (c) areselected from the group consisting of volume per mole, volume perhydrogen atom, and volume per carbon atom.
 13. The method of claim 9,wherein (e) further comprises determining the distribution of chainlengths of the constituents of said sample under investigation.
 14. Amethod for determining the characteristics of a fluid sample,comprising: a. obtaining measurements of a plurality of calibrationsamples at more than one temperature and a reference pressure, whereinthe calibration samples have differing mean chain lengths, and whereinsaid measurements are selected from the group consisting of diffusionmeasurements and relaxation measurements; b. obtaining measurements ofsaid sample under investigation at a temperature within or near therange of temperatures of the measurements of (a) and at a secondpressure, wherein said measurements are selected from the groupconsisting of diffusion measurements and relaxation measurements; c.determining the relationship of volume of one or more alkanes ormixtures of alkanes to (i) the mean chain length at the referencepressure and (ii) the mean chain length at the second pressure; d.determining the scaling law in terms chain length at the second pressureusing the functions of (c) and the measurements of (a); e. applying thescaling law of (d) with the measurements of (b) to determine thecomposition of the sample under investigation.
 15. The method of claim14, wherein the scaling law of (d) is a function of chain length andmean chain length.
 16. The method of claim 14, wherein said volumefunctions of (c) are obtained using standard look-up tables.
 17. Themethod of claim 14, wherein the volumes of (c) are selected from thegroup consisting of volume per mole, volume per hydrogen atom, andvolume per carbon atom.
 18. The method of claim 14, wherein (e) furthercomprises determining the distribution of chain lengths of theconstituents of said sample under investigation.